A straight highway leads to the top of the base of a 50 meter tall tower. Frm the top of the tower, if yhe angles of depression of two cars on the road are 30° and 60° , then the distence between the tow cars is ................meter.
We form 2 rt. triangles with a common ver. side:
Y = 50 m. = Ver. side.
X1 = hor. side of larger triangle(30 Deg).
X2 = hor. side of smaller triangle(60 Deg).
tan30 = Y/x1 = 50/X1.
X1 = 50/tan30 = 86.6 m.
X2 = 50/tan60 = 28.9 m.
d = X1 - X2 = 86.6 - 28.9 = 57.7 m. =
The distance between cars.
To solve this problem, we need to use trigonometric ratios and the concept of similar triangles. Let's break it down step by step:
1. Draw a diagram: Draw a rough diagram of the situation. Place the tower at the origin and the two cars on the road at some distance away from the tower.
2. Identify the angles of depression: One car has an angle of depression of 30°, and the other car has an angle of depression of 60°. The angles of depression are measured from the top of the tower to each car.
3. Draw lines: Draw lines from the top of the tower to each car. This creates two right-angled triangles.
4. Find the height of the tower: The tower's height is given as 50 meters.
5. Calculate distances: Let's assume that the distance from the tower to the first car is x meters. Since we have a right-angled triangle, we can use trigonometric ratios to solve for x.
- In the first triangle, tan(30°) = height of the tower / x. Therefore, x = height of the tower / tan(30°).
- In the second triangle, tan(60°) = height of the tower / (x + distance between the two cars). Therefore, distance between the two cars = (height of the tower / tan(60°)) - x.
6. Substitute values and solve: Substitute the value of the height of the tower (50 meters) into the equations for x and the distance between the two cars. Then solve for x and the distance.
- x = 50 meters / tan(30°)
- distance between the two cars = (50 meters / tan(60°)) - x
7. Calculate the final result: Plug in the values and calculate the distance between the two cars.
It is important to note that all calculations should be done in radians for accuracy.
By following these steps, you should be able to find the distance between the two cars.